Question

Integrate the following functions.
$\int x\sqrt{1+2x^2}dx.$

Answer 1

Let $I=\int x\sqrt{1+2x^2}dx$
Let $1+2x^2=t$
On differentiating w.r.t.x, we get $4x=\frac{dt}{dx}\Rightarrow dx=\frac{dt}{4x}$
$\therefore I=\int x\sqrt{t}\frac{dt}{4x}=\frac{1}{4}\int \sqrt{t}dt=\frac{1}{4}\int t^{\frac{1}{2}}dt=\frac{1}{4}\frac{t^{\frac{1}{2}}+1}{\left(\frac{1}{2}\right)+1}+C=\frac{1}{6}(1+2x^2{)}^{\frac{3}{2}}+C$

Note In the given type of integrals, it is better to substitute the root function so that it may be converted into a standard integral.